## Modular Value Distribution

Goal:

In many cases one is interested in the values taken by a function. In this project we study the values of integer polynomials modulo integers. For instance, the values of f(x)=x^2 at natural numbers are precisely the squares: 0,1,4,16,25,36,... Considering them modulo 8, one finds 0,1,4,0,1,4,... In fact, one easily proves that 0,1,4 are the only residues modulo 8 that appear for squares. One could consider other polynomials like f(x)=x^3 or f(x)=x^2+x+1. The modulus can be a prime, a prime power or any positive integer. In fact, the latter case can usually be reduced to the previous ones by the Chinese Remainder Theorem (see Algebra lecture).

The aim is to make computer experimentation on this kind of question, to nicely present and illustrate the results, and to see what can be theoretically explained by the Chinese Remainder Theorem.

This project is motivated by a concrete question from a research project. The questions and the details will be explained if the project is chosen.

Schedule: To be determined.

Supervisors: Gabor Wiese

Difficulty level: Introductory.

Tools: Any computer language.

Results: [too be completed at the end of the project.]